A Monte Carlo model is developed for predicting electrical conductivity of carbon nanofiber composite materials. The conductive nanofibers are models as both 2D and 3D network of finite sites that are randomly distributed. The percolation behavior of the network is studied using the Monte Carlo method, which leads to the determination of the percolation threshold. The effect of the nanofiber aspect ratio on the critical nanofiber volume rate is investigated in the current model, each of the nanofibers needs five independent geometrical parameters (i.e., three coordinates in space and two orientation angles) for its identification. There are three controlling parameters for each nanofiber, which includes the nanofiber length, the nanofiber diameter, and the nanofiber aspect ratio. The simulation results reveal a relationship between the fiber aspect ratio and the percolation threshold: the higher the aspect ratio, the lower the threshold. With the simulation results obtained from the Monte Carlo model, the effective electrical conductivity of the composite is then determined by assuming the conductivity is proportional to the ratio of the number of nanofibers forming the largest cluster to the total number of nanofibers. The numerical results indicate that as the volume rate reaches a critical value, the conductivity starts to rise sharply. These obtained simulation results agree fairly with experimental and numerical data published earlier by others. In addition, we investigate the convergence of the current percolation model. We also find the tunneling effect does not affect the critical volume rate greatly. We propose that the percolation model is not scalable as well.

While the statistical mechanics of systems in thermal equilibrium is a well established discipline, nonequilibrium systems are fundamentally much less well understood, even though most natural phenomena fall into the latter category. In particular, there is as yet no nonequilibrium analog for the systematic formalism of Gibbs ensembles. Rather than deal with the difficult problem of general nonequilibrium systems, this study is restricted to the steady states of a simple model whose equilibrium properties are well known. The Ising lattice gas displays a number of surprising phenomena when driven into nonequilibrium steady states. This study extends previous work to a more general model with anisotropic interparticle interactions. Using Monte Carlo simulations, we obtain the phase diagram for the model, controlled by the driving field, temperature, and anisotropy parameter α. Under saturation drive, the shift in the transition temperature between ordered and disordered states can be either positive or negative, depending on α ≡ √(𝐽_∥/𝐽_⟂). The possible existence at large α of an additional phase ordered in only one direction is discussed. For finite drives, both first and second order transitions are observed. A novel technique for locating the first order transition line is presented. Some aspects of the phase diagram can be predicted by investigating the two-point correlation function to first order in a high temperature series expansion. However, the series expansion fails to predict even qualitatively the α-dependence of the critical temperature. / Master of Science

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Page iii not included in the thesis and therefore, not included in the page count. / <p> In recent years, some efficient methods have been developed for calculating derivative price sensitivities, or the Greeks, using Monte Carlo simulation. However, the slow convergence, especially for discontinuous payoff functions, is well known for Monte Carlo simulation. In this project, we investigate the Malliavin calculus and its application in computation of the Greeks. Malliavin calculus and Wiener Chaos theory are introduced. The theoretical framework of the Malliavin weighted scheme of computation of the Greeks is explored in details, and the numerical implementation of the one-dimensional case and an example of the two-dimensional case are presented. Finally, the results are compared with those of finite difference scheme.</p> / Thesis / Master of Science (MSc)

We study the phase diagram of an interacting lattice gas of two species of particles and holes, driven out of equilibrium by a local hopping bias (denoted by `E'). Particles interact by excluded volume and nearest-neighbor attractions. We present a detailed Monte Carlo investigation of the phase diagram. Three phases are found, with a homogenous phase at high temperatures and two distinct ordered phases at lower temperatures. Which ordered phase is observed depends on the parameter f, which controls the ratio of the two types of particles. At small f, there is nearly a single species, and a transition is observed into a KLS-type ordered phase. At larger f, the minority species are sufficiently dense to form a transverse blockage, and a sequence of two transitions are observed as the temperature is lowered. First, a continuous boundary is crossed into an SHZ-type ordered phase, then at a lower temperature a first-order boundary is crossed into the KLS-type ordered phase. At some critical value of f is a bicritical point, where the first-order line branches from the two continuous boundaries. We also consider correlations in the homogenous phase, by constructing a continuum description and comparing to the results of simulations. Long range correlations are present in both the theoretical results and the simulations, though certain details of the theory do not fit the observations very well. Finally, we examine the beahvior of three-point correlations in the single-species (KLS) limit. Nontrivial three-point correlations are directly related to the nonzero bias E. We therefore consider the behavior of the three-point correlations as a function of E. We find that the three-point signal saturates very rapidly with E. There are some difficulties interpreting the data at small E. / Ph. D.